# Ince Polynomials

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An Ince polynomial [1] of order and degree is a solution of the periodic linear second-order differential equation . Ince polynomials are divided into two families and [2]. You can select either an analytic form or a plot of the function. Ince polynomials can also be represented by trigonometric series. These functions are of importance in applications including beam optics.

Contributed by: Enrique Zeleny (April 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The recurrence relations for the solutions were taken from [2].

References

[1] "Mathieu Functions and Hill's Equation." NIST Digital Library of Mathematical Functions. (Apr 4, 2014) dlmf.nist.gov/28.31.

[2] M. A. Bandres and J. C. Gutiérrez-Vega, "Ince–Gaussian Modes of the Paraxial Wave Equation and Stable Resonators," *Journal of the Optical Society of America A—Optics, Image Science, and Vision*, 21(5), 2004 pp. 873–880. doi:10.1364/JOSAA.21.000873.

## Permanent Citation

"Ince Polynomials"

http://demonstrations.wolfram.com/IncePolynomials/

Wolfram Demonstrations Project

Published: April 7 2014